Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,4,Mod(79,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.79");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.j (of order \(6\), degree \(2\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(14.4554679514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 55 x^{18} + 2042 x^{16} - 41247 x^{14} + 600234 x^{12} - 4812047 x^{10} + 27547801 x^{8} + \cdots + 12960000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2}\cdot 7^{8} \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{20} - 55 x^{18} + 2042 x^{16} - 41247 x^{14} + 600234 x^{12} - 4812047 x^{10} + 27547801 x^{8} + \cdots + 12960000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 23\!\cdots\!19 \nu^{18} + \cdots - 14\!\cdots\!00 ) / 92\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 23\!\cdots\!57 \nu^{18} + \cdots - 87\!\cdots\!00 ) / 92\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 82\!\cdots\!27 \nu^{18} + \cdots - 37\!\cdots\!00 ) / 73\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\!\cdots\!13 \nu^{18} + \cdots + 99\!\cdots\!16 ) / 11\!\cdots\!64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22\!\cdots\!01 \nu^{18} + \cdots - 15\!\cdots\!00 ) / 14\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 25\!\cdots\!59 \nu^{18} + \cdots - 11\!\cdots\!00 ) / 14\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 51\!\cdots\!89 \nu^{19} + \cdots - 30\!\cdots\!00 \nu ) / 11\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 16\!\cdots\!31 \nu^{18} + \cdots - 81\!\cdots\!00 ) / 73\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 66\!\cdots\!21 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 66\!\cdots\!21 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 25\!\cdots\!43 \nu^{19} + \cdots + 39\!\cdots\!00 \nu ) / 44\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 76\!\cdots\!49 \nu^{19} + \cdots + 35\!\cdots\!00 \nu ) / 11\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 73\!\cdots\!71 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 88\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 73\!\cdots\!71 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 88\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 74\!\cdots\!53 \nu^{19} + \cdots + 19\!\cdots\!00 \nu ) / 63\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 84\!\cdots\!53 \nu^{19} + \cdots - 22\!\cdots\!00 \nu ) / 44\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 88\!\cdots\!43 \nu^{19} + \cdots + 46\!\cdots\!00 \nu ) / 44\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 33\!\cdots\!11 \nu^{19} + \cdots + 16\!\cdots\!00 \nu ) / 44\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 53\!\cdots\!33 \nu^{19} + \cdots - 26\!\cdots\!00 \nu ) / 44\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{15} + \beta_{11} - 13\beta_{7} ) / 14 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} + 2 \beta_{5} + \cdots + 81 \beta_1 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 8 \beta_{19} + 25 \beta_{18} - 61 \beta_{17} + 8 \beta_{16} - 25 \beta_{15} - 4 \beta_{14} + \cdots - 241 \beta_{7} ) / 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -48\beta_{10} - 48\beta_{9} - 115\beta_{8} + 10\beta_{6} + 70\beta_{5} + 10\beta_{3} + 1581\beta _1 - 1581 ) / 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 204 \beta_{19} + 609 \beta_{18} - 1793 \beta_{17} - 228 \beta_{14} + 228 \beta_{13} + \cdots - 228 \beta_{9} ) / 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1330\beta_{14} + 1330\beta_{13} + 400\beta_{6} - 1986\beta_{4} - 2425\beta_{2} - 35201 ) / 7 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -4536\beta_{16} + 15737\beta_{15} - 48077\beta_{11} + 8036\beta_{10} - 8036\beta_{9} + 120609\beta_{7} ) / 14 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 36864 \beta_{14} + 36864 \beta_{13} + 36864 \beta_{10} + 36864 \beta_{9} + 50875 \beta_{8} + \cdots - 823061 \beta_1 ) / 7 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 99020 \beta_{19} - 411393 \beta_{18} + 1260785 \beta_{17} - 99020 \beta_{16} + 411393 \beta_{15} + \cdots + 2841353 \beta_{7} ) / 14 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 1000642 \beta_{10} + 1000642 \beta_{9} + 1083585 \beta_{8} - 346840 \beta_{6} - 1417162 \beta_{5} + \cdots + 19725241 ) / 7 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 2186168 \beta_{19} - 10724825 \beta_{18} + 32751261 \beta_{17} + 6912884 \beta_{14} + \cdots + 6912884 \beta_{9} ) / 14 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 26678288 \beta_{14} - 26678288 \beta_{13} - 9423130 \beta_{6} + 37043350 \beta_{4} + 23587475 \beta_{2} + 480275741 ) / 7 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 49236364 \beta_{16} - 278156449 \beta_{15} + 845999713 \beta_{11} - 189045508 \beta_{10} + \cdots - 1659785417 \beta_{7} ) / 14 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 702074130 \beta_{14} - 702074130 \beta_{13} - 702074130 \beta_{10} - 702074130 \beta_{9} + \cdots + 11828617841 \beta_1 ) / 7 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 1132842296 \beta_{19} + 7181518137 \beta_{18} - 21770396397 \beta_{17} + \cdots - 40854349729 \beta_{7} ) / 14 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 18307346784 \beta_{10} - 18307346784 \beta_{9} - 11992550315 \beta_{8} + 6583635570 \beta_{6} + \cdots - 293848013861 ) / 7 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 26591749580 \beta_{19} + 184769275713 \beta_{18} - 558733748305 \beta_{17} + \cdots - 133280657652 \beta_{9} ) / 14 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 474310356002 \beta_{14} + 474310356002 \beta_{13} + 171388869800 \beta_{6} - 638259667162 \beta_{4} + \cdots - 7347685400361 ) / 7 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 635230140728 \beta_{16} + 4741634904985 \beta_{15} - 14312500385981 \beta_{11} + \cdots + 25351742981601 \beta_{7} ) / 14 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−4.60176 | − | 2.65683i | 1.67956 | − | 0.969693i | 10.1175 | + | 17.5240i | 8.30643 | + | 7.48353i | −10.3052 | 0 | − | 65.0123i | −11.6194 | + | 20.1254i | −18.3418 | − | 56.5062i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.2 | −3.50611 | − | 2.02426i | −5.65297 | + | 3.26374i | 4.19523 | + | 7.26634i | −5.00915 | − | 9.99542i | 26.4266 | 0 | − | 1.58074i | 7.80403 | − | 13.5170i | −2.67065 | + | 45.1849i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.3 | −2.47228 | − | 1.42737i | 7.78434 | − | 4.49429i | 0.0747741 | + | 0.129513i | −7.11340 | − | 8.62551i | −25.6601 | 0 | 22.4110i | 26.8973 | − | 46.5874i | 5.27451 | + | 31.4781i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.4 | −1.45073 | − | 0.837581i | 2.15983 | − | 1.24698i | −2.59692 | − | 4.49799i | 11.1437 | − | 0.904354i | −4.17779 | 0 | 22.1018i | −10.3901 | + | 17.9961i | −16.9240 | − | 8.02178i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.5 | −1.34691 | − | 0.777638i | −4.29677 | + | 2.48074i | −2.79056 | − | 4.83339i | −4.33318 | + | 10.3065i | 7.71648 | 0 | 21.1224i | −1.19182 | + | 2.06430i | 13.8511 | − | 10.5122i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.6 | 1.34691 | + | 0.777638i | 4.29677 | − | 2.48074i | −2.79056 | − | 4.83339i | −6.75908 | + | 8.90588i | 7.71648 | 0 | − | 21.1224i | −1.19182 | + | 2.06430i | −16.0294 | + | 6.73929i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.7 | 1.45073 | + | 0.837581i | −2.15983 | + | 1.24698i | −2.59692 | − | 4.49799i | −4.78866 | − | 10.1029i | −4.17779 | 0 | − | 22.1018i | −10.3901 | + | 17.9961i | 1.51494 | − | 18.6675i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.8 | 2.47228 | + | 1.42737i | −7.78434 | + | 4.49429i | 0.0747741 | + | 0.129513i | 11.0266 | + | 1.84763i | −25.6601 | 0 | − | 22.4110i | 26.8973 | − | 46.5874i | 24.6236 | + | 20.3069i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.9 | 3.50611 | + | 2.02426i | 5.65297 | − | 3.26374i | 4.19523 | + | 7.26634i | 11.1609 | − | 0.659663i | 26.4266 | 0 | 1.58074i | 7.80403 | − | 13.5170i | 40.4666 | + | 20.2796i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.10 | 4.60176 | + | 2.65683i | −1.67956 | + | 0.969693i | 10.1175 | + | 17.5240i | −10.6341 | − | 3.45182i | −10.3052 | 0 | 65.0123i | −11.6194 | + | 20.1254i | −39.7649 | − | 44.1375i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.1 | −4.60176 | + | 2.65683i | 1.67956 | + | 0.969693i | 10.1175 | − | 17.5240i | 8.30643 | − | 7.48353i | −10.3052 | 0 | 65.0123i | −11.6194 | − | 20.1254i | −18.3418 | + | 56.5062i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.2 | −3.50611 | + | 2.02426i | −5.65297 | − | 3.26374i | 4.19523 | − | 7.26634i | −5.00915 | + | 9.99542i | 26.4266 | 0 | 1.58074i | 7.80403 | + | 13.5170i | −2.67065 | − | 45.1849i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.3 | −2.47228 | + | 1.42737i | 7.78434 | + | 4.49429i | 0.0747741 | − | 0.129513i | −7.11340 | + | 8.62551i | −25.6601 | 0 | − | 22.4110i | 26.8973 | + | 46.5874i | 5.27451 | − | 31.4781i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.4 | −1.45073 | + | 0.837581i | 2.15983 | + | 1.24698i | −2.59692 | + | 4.49799i | 11.1437 | + | 0.904354i | −4.17779 | 0 | − | 22.1018i | −10.3901 | − | 17.9961i | −16.9240 | + | 8.02178i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.5 | −1.34691 | + | 0.777638i | −4.29677 | − | 2.48074i | −2.79056 | + | 4.83339i | −4.33318 | − | 10.3065i | 7.71648 | 0 | − | 21.1224i | −1.19182 | − | 2.06430i | 13.8511 | + | 10.5122i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.6 | 1.34691 | − | 0.777638i | 4.29677 | + | 2.48074i | −2.79056 | + | 4.83339i | −6.75908 | − | 8.90588i | 7.71648 | 0 | 21.1224i | −1.19182 | − | 2.06430i | −16.0294 | − | 6.73929i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.7 | 1.45073 | − | 0.837581i | −2.15983 | − | 1.24698i | −2.59692 | + | 4.49799i | −4.78866 | + | 10.1029i | −4.17779 | 0 | 22.1018i | −10.3901 | − | 17.9961i | 1.51494 | + | 18.6675i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.8 | 2.47228 | − | 1.42737i | −7.78434 | − | 4.49429i | 0.0747741 | − | 0.129513i | 11.0266 | − | 1.84763i | −25.6601 | 0 | 22.4110i | 26.8973 | + | 46.5874i | 24.6236 | − | 20.3069i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.9 | 3.50611 | − | 2.02426i | 5.65297 | + | 3.26374i | 4.19523 | − | 7.26634i | 11.1609 | + | 0.659663i | 26.4266 | 0 | − | 1.58074i | 7.80403 | + | 13.5170i | 40.4666 | − | 20.2796i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 214.10 | 4.60176 | − | 2.65683i | −1.67956 | − | 0.969693i | 10.1175 | − | 17.5240i | −10.6341 | + | 3.45182i | −10.3052 | 0 | − | 65.0123i | −11.6194 | − | 20.1254i | −39.7649 | + | 44.1375i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 35.j | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.4.j.f | 20 | |
| 5.b | even | 2 | 1 | inner | 245.4.j.f | 20 | |
| 7.b | odd | 2 | 1 | 245.4.j.e | 20 | ||
| 7.c | even | 3 | 1 | 245.4.b.d | 10 | ||
| 7.c | even | 3 | 1 | inner | 245.4.j.f | 20 | |
| 7.d | odd | 6 | 1 | 35.4.b.a | ✓ | 10 | |
| 7.d | odd | 6 | 1 | 245.4.j.e | 20 | ||
| 21.g | even | 6 | 1 | 315.4.d.c | 10 | ||
| 28.f | even | 6 | 1 | 560.4.g.f | 10 | ||
| 35.c | odd | 2 | 1 | 245.4.j.e | 20 | ||
| 35.i | odd | 6 | 1 | 35.4.b.a | ✓ | 10 | |
| 35.i | odd | 6 | 1 | 245.4.j.e | 20 | ||
| 35.j | even | 6 | 1 | 245.4.b.d | 10 | ||
| 35.j | even | 6 | 1 | inner | 245.4.j.f | 20 | |
| 35.k | even | 12 | 1 | 175.4.a.i | 5 | ||
| 35.k | even | 12 | 1 | 175.4.a.j | 5 | ||
| 35.l | odd | 12 | 1 | 1225.4.a.be | 5 | ||
| 35.l | odd | 12 | 1 | 1225.4.a.bh | 5 | ||
| 105.p | even | 6 | 1 | 315.4.d.c | 10 | ||
| 105.w | odd | 12 | 1 | 1575.4.a.bn | 5 | ||
| 105.w | odd | 12 | 1 | 1575.4.a.bq | 5 | ||
| 140.s | even | 6 | 1 | 560.4.g.f | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.b.a | ✓ | 10 | 7.d | odd | 6 | 1 | |
| 35.4.b.a | ✓ | 10 | 35.i | odd | 6 | 1 | |
| 175.4.a.i | 5 | 35.k | even | 12 | 1 | ||
| 175.4.a.j | 5 | 35.k | even | 12 | 1 | ||
| 245.4.b.d | 10 | 7.c | even | 3 | 1 | ||
| 245.4.b.d | 10 | 35.j | even | 6 | 1 | ||
| 245.4.j.e | 20 | 7.b | odd | 2 | 1 | ||
| 245.4.j.e | 20 | 7.d | odd | 6 | 1 | ||
| 245.4.j.e | 20 | 35.c | odd | 2 | 1 | ||
| 245.4.j.e | 20 | 35.i | odd | 6 | 1 | ||
| 245.4.j.f | 20 | 1.a | even | 1 | 1 | trivial | |
| 245.4.j.f | 20 | 5.b | even | 2 | 1 | inner | |
| 245.4.j.f | 20 | 7.c | even | 3 | 1 | inner | |
| 245.4.j.f | 20 | 35.j | even | 6 | 1 | inner | |
| 315.4.d.c | 10 | 21.g | even | 6 | 1 | ||
| 315.4.d.c | 10 | 105.p | even | 6 | 1 | ||
| 560.4.g.f | 10 | 28.f | even | 6 | 1 | ||
| 560.4.g.f | 10 | 140.s | even | 6 | 1 | ||
| 1225.4.a.be | 5 | 35.l | odd | 12 | 1 | ||
| 1225.4.a.bh | 5 | 35.l | odd | 12 | 1 | ||
| 1575.4.a.bn | 5 | 105.w | odd | 12 | 1 | ||
| 1575.4.a.bq | 5 | 105.w | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{20} - 58 T_{2}^{18} + 2255 T_{2}^{16} - 47426 T_{2}^{14} + 714581 T_{2}^{12} - 6457776 T_{2}^{10} + \cdots + 655360000 \)
|
|
\( T_{19}^{10} - 36 T_{19}^{9} + 18646 T_{19}^{8} - 493432 T_{19}^{7} + 295286596 T_{19}^{6} + \cdots + 405342520934400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 655360000 \)
|
| $3$ |
\( T^{20} + \cdots + 3930163511296 \)
|
| $5$ |
\( T^{20} + \cdots + 93\!\cdots\!25 \)
|
| $7$ |
\( T^{20} \)
|
| $11$ |
\( (T^{10} + \cdots + 124078925021184)^{2} \)
|
| $13$ |
\( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 16\!\cdots\!96 \)
|
| $19$ |
\( (T^{10} + \cdots + 405342520934400)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 58\!\cdots\!96 \)
|
| $29$ |
\( (T^{5} - 44 T^{4} + \cdots - 126081243400)^{4} \)
|
| $31$ |
\( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 35\!\cdots\!96 \)
|
| $41$ |
\( (T^{5} - 426 T^{4} + \cdots - 109849343072)^{4} \)
|
| $43$ |
\( (T^{10} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 11\!\cdots\!96 \)
|
| $53$ |
\( T^{20} + \cdots + 30\!\cdots\!36 \)
|
| $59$ |
\( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{10} + \cdots + 42\!\cdots\!64)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 63\!\cdots\!76 \)
|
| $71$ |
\( (T^{5} + \cdots - 7767441797120)^{4} \)
|
| $73$ |
\( T^{20} + \cdots + 78\!\cdots\!96 \)
|
| $79$ |
\( (T^{10} + \cdots + 40\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots + 78\!\cdots\!96)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 15\!\cdots\!64)^{2} \)
|
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